1. Field of the Invention
Equations for determining electrostatic potential in the study of electromagnetism, or for determining pressure via Navier-Stokes equations in the study of incompressible fluid hydrodynamics, are collectively known by the form of Poisson equations. Further, properties of heat conduction or material diffusion can be defined by and follow equations known as diffusion equations. The present invention relates to a computational method for numerically calculating such Poisson equations, diffusion equations, or similar kinds of partial differential equations by means of a program for performing numerical calculation for simulation of physical phenomena or a program for performing design evaluation using such equations.
2. Description of the Related Art
Conventional known programs for calculating Poisson equations or diffusion equations have employed only the dependent variables of the equations per se. When the grid points on which such equations are operated are spaced at uniform intervals, a calculation having second order accuracy can be performed using center difference approximation. However, in the case of irregular intervals, computational accuracy with this known method has proven insufficient. Examples of a calculations involving the center difference approximation method are disclosed by C. A. J. Fletcher, xe2x80x9cComputational Techniques for Fluid Dynamics,xe2x80x9d I, II, Springer-Verlag (1988).
Poisson equations or diffusion equations are calculated on the basis of certain given boundary conditions. In the case that the spatial geometry of shape to be defined by such boundary conditions is complex, or when irregular interval grid points are used, a problem arises in that boundary conditions of sufficient accuracy cannot be derived. Further, for equations in which a transform of space coordinates which are made to follow a given boundary is performed, wherein such equations employ regular interval grid points based on such transformed coordinates, i.e. boundary-fitted coordinates, it is frequently the case that considerable effort is necessary for implementing such equations in the case of a complex geometry, and the fact remains that extremely difficult cases are encountered which have resulted in major problems.
In the present invention, in order to solve with good accuracy a Poisson equation, diffusion equation or like partial differential equation performed on a plurality of grid points dispersed at irregular intervals, not only the values of the dependent variable of the original equations are used, but rather a program is created in which the first order derivatives thereof also are introduced independently as dependent variables, and discretized equations are generated by using discretized expressions for each of second and third order derivatives of the dependent variable of the original equation.